cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Les Reid

Les Reid's wiki page.

Les Reid has authored 4 sequences.

A213323 Number of permutations of n objects such that no four-element subset is preserved.

Original entry on oeis.org

1, 1, 2, 6, 0, 44, 304, 2568, 26704, 200240, 1931616, 20849696, 246556672, 3300906816, 46382446720, 695413794944, 11120648673024, 188600719094528, 3394592207824384, 64513420630110720, 1290420198709682176, 27102196040419214336, 596237419436696543232, 13713106494042086045696
Offset: 0

Views

Author

Les Reid, Jun 08 2012

Keywords

Comments

The limit as n -> infinity of a(n)/n! = (13+9*exp(1/3))/(6*exp(25/12)) or approximately 0.5304422700.

Examples

			Example: For n=5 the only permutations that fix no four-element subset are the 24 5-cycles and the 20 products of a 3-cycle and a 2-cycle, therefore a(5)=44.

Crossrefs

Cf. A000166, A137482, A213322, A213324

Programs

  • PARI
    x='x+O('x^66);
egf=((x+x^2/2+2*x^3/3)*exp(-x-x^2/2-x^3/3-x^4/4)+(1+x^2/2)*exp(-x-x^2/2-x^4/4))/(1-x);
Vec(serlaplace(egf))
/* Joerg Arndt, Jun 09 2012 */

Formula

E.g.f.: ((x+x^2/2+2*x^3/3)*exp(-x-x^2/2-x^3/3-x^4/4)+(1+x^2/2)*exp(-x-x^2/2-x^4/4))/(1-x)

A213324 Number of permutations of n objects such that no five-element subset is preserved.

Original entry on oeis.org

1, 1, 2, 6, 24, 0, 265, 2260, 20145, 200240, 2492225, 23163480, 270877705, 3449462080, 48030998625, 713129276000, 11685451112225, 198919432944000, 3585292622812225, 68053546078588000, 1360638669122771625, 28525836193802883200, 627637954389517169825, 14435957818250131813200, 346518764145610187160625
Offset: 0

Views

Author

Les Reid, Jun 08 2012

Keywords

Comments

Limit_{n->oo} a(n)/n! = (35-24*exp(1/4)+24*exp(1/3)+24*exp(7/12)+24*exp(3/4))/(24*exp(137/60)) = 0.5585422951...

Examples

			For n=6 the only permutations that fix no five-element subset are the 120 6-cycles, the 90 products of a 4-cycle and a 2-cycle, the 40 products of two 3-cycles, and the 15 products of three 2-cycles, therefore a(5)=265.

Crossrefs

Cf. A000166, A137482, A213322, A213323

Programs

  • PARI
    x='x+O('x^66);
egf=((x^2/2+2*x^3/3+7*x^4/24)*exp(-x-x^2/2-x^3/3-x^4/4-x^5/5)+x*exp(-x-x^2/2-x^4/4-x^5/5)+exp(-x-x^2/2-x^5/5)+exp(-x-x^3/3-x^5/5)-exp(-x-x^2/2-x^3/3-x^5/5))/(1-x);
Vec(serlaplace(egf))
/* Joerg Arndt, Jun 09 2012 */

Formula

E.g.f.: ((x^2/2+2*x^3/3+7*x^4/24)*exp(-x-x^2/2-x^3/3-x^4/4-x^5/5)+x*exp(-x-x^2/2-x^4/4-x^5/5)+exp(-x-x^2/2-x^5/5)+exp(-x-x^3/3-x^5/5)-exp(-x-x^2/2-x^3/3-x^5/5))/(1-x).

A213322 Number of permutations of n objects such that no three-element subset is preserved.

Original entry on oeis.org

1, 1, 2, 0, 9, 54, 459, 2568, 20145, 176076, 1833741, 20148336, 241870617, 3132196560, 43874128089, 658195206264, 10533823597089, 179062417518768, 3223079582143185, 61237777946016096, 1224762717659002281, 25720036368344942616, 565841009719801635777
Offset: 0

Views

Author

Les Reid, Jun 08 2012

Keywords

Comments

The limit as n -> infinity of a(n)/n! = (3+2*exp(1/2))/(2*exp(11/6)) or approximately 0.5034167572.

Examples

			Example: For n=5 the only permutations that fix no three-element subset are the 24 5-cycles and the 30 4-cycles, therefore a(5)=54.

Crossrefs

Cf. A000166, A137482, A213323, A213324.

Programs

  • PARI
    lista(nn) = {x=xx+O(xx^nn); egf=((x+x^2/2)*exp(-x-x^2/2-x^3/3)+exp(-x-x^3/3))/(1-x); Vec(serlaplace(egf)) ;} \\ Michel Marcus, Aug 14 2013

Formula

E.g.f.:((x+x^2/2)*exp(-x-x^2/2-x^3/3)+exp(-x-x^3/3))/(1-x)

Extensions

More terms from Michel Marcus, Aug 14 2013

A066951 Number of nonisomorphic connected graphs that can be drawn in the plane using n unit-length edges.

Original entry on oeis.org

1, 1, 3, 5, 12, 28, 74, 207, 633, 2008
Offset: 1

Views

Author

Les Reid, May 25 2002

Keywords

Comments

K_4 can't be so drawn even though it is planar. These graphs are a subset of those counted in A046091.

Examples

			Up to five edges, every planar graph can be drawn with edges of length 1, so up to this point the sequence agrees with A046091 (connected planar graphs with n edges) [except for the fact that that sequence begins with no edges]. For six edges, the only graphs that cannot be drawn with edges of length 1 are K_4 and K_{3,2}. According to A046091, there are 30 connected planar graphs with 6 edges, so the sixth term is 28.

References

  • M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 80.
  • R. C. Read, From Forests to Matches, Journal of Recreational Mathematics, Vol. 1:3 (Jul 1968), 60-172.

Links

Crossrefs

Cf. A003055, A002905, A046091.

Extensions

a(7) = 70. - Jonathan Vos Post, Jan 05 2007
Corrected, extended and reference added. a(7)=74 and a(8)=207 from Read's paper. - William Rex Marshall, Nov 16 2010
a(9) from Salvia's paper added by Brendan McKay, Apr 13 2013
a(9) corrected (from version 2 [May 22 2013] of Salvia's paper) by Gaetano Ricci, May 24 2013
a(10) from Vaisse's webpage added by Raffaele Salvia, Jan 31 2015

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.